how to pass quantitiave and reasoning tests
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C H A P T E R 1
Review the basics
Chapter topics
Terms used in this chapterL
Multiplication tablesL
Dividing and multiplying numbersL
Prime numbersL
MultiplesL
Working with large numbersL
Working with signed numbersL
AveragesL
Answers to Chapter 1L
Terms used in this chapterArithmetic mean: The amount obtained by adding two or more
numbers and dividing by the number of terms.
9
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HOW TO PASS NUMERICAL REASONING TESTS10
Average: See Mode, Median and Arithmetic mean.
Dividend: The number to be divided.
Divisor: The number by which another is divided.
Factor: The positive integers by which an integer is evenly divisible.
Find the product of …: Multiply two or more numbers together.
Integer: A whole number without decimal or fraction parts.
Lowest common multiple: The least quantity that is a multiple of
two or more given values.
Mean: See Arithmetic mean.
Median: The middle number in a range of numbers when the set is
arranged in ascending or descending order.
Mode: The most popular value in a set of numbers.
Multiple: A number that divides into another without a remainder.
Prime factor: The factors of an integer that are prime numbers.
Prime number: A number divisible only by itself and 1.
Test-writers assume that you remember the fundamentals you learnt
in school and that you can apply that knowledge and understanding
to the problems in the tests. The purpose of this chapter is to remind
you of the basics and to provide you with the opportunity to practise
them before your test. The skills you will learn in this chapter are the
fundamentals you can apply to solving many of the problems in an
aptitude test, so it is worth learning the basics thoroughly. You mustbe able to do simple calculations very quickly, without expending
any unnecessary brainpower keep this in reserve for the tricky
questions later on. This chapter reviews the basics and includes a
number of practice drills to ease you back into numerical shape.
Remember, no calculators …
Multiplication tables
‘Rote learning’ as a teaching method has fallen out of favour in
recent years. There are good reasons for this in some academicareas but it doesn’t apply to multiplication tables. You learnt the
times-tables when you first went to school, but can you recite the
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REVIEW THE BASICS 11
tables as quickly now? Recite them to yourself quickly, over and
over again, when you’re out for a run, when you’re washing up,
when you’re cleaning your teeth, when you’re stirring your baked
beans any time when you have a spare 10 seconds thinking time.
Six times, seven times and eight times are the easiest to forget, so
drill these more often than the twos and fives. Make sure that you
can respond to any multiplication question without pausing even
for half a second. If you know the multiplication tables inside out,
you will save yourself valuable seconds in your test and avoid need-
less mistakes in your calculations.
Multiplication tables: practice drill1
Practise these drills and aim to complete each set within 15 sec-
onds. (Remember, the answers are at the end of the chapter.)
Drill 1 Drill 2 Drill 3 Drill 4 Drill 5
1 3 s 7 8 s 3 9 s 3 7 s 5 7 s 15
2 6 s 5 11 s 6 9 s 2 6 s 3 11 s 11
3 8 s 9 13 s 2 7 s 7 4 s 7 4 s 12
4 3 s 3 11 s 13 12 s 7 3 s 4 8 s 10
5 9 s 12 13 s 9 6 s 8 8 s 15 13 s 4
6 2 s 4 6 s 14 6 s 7 8 s 8 11 s 2
7 8 s 5 3 s 15 13 s 5 2 s 6 7 s 12
8 13 s 3 9 s 8 13 s 4 7 s 6 9 s 15
9 6 s 7 4 s 5 8 s 8 4 s 12 9 s 9
10 2 s 7 6 s 3 7 s 13 5 s 14 3 s 8
11 7 s 12 12 s 4 6 s 15 3 s 11 3 s 3
12 2 s 12 11 s 7 9 s 8 9 s 6 3 s 8
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HOW TO PASS NUMERICAL REASONING TESTS12
Multiplication tables: practice drill 2
Drill 1 Drill 2 Drill 3 Drill 4 Drill 5
1 8 × 9 4 × 5 3 × 6 6 × 11 11 × 11
2 13 × 13 9 × 3 9 × 5 5 × 3 3 × 7
3 11 × 14 13 × 4 7 × 3 5 × 6 6 × 8
4 9 × 6 6 × 8 5 × 9 8 × 5 12 × 4
5 11 × 5 7 × 7 11 × 15 14 × 14 5 × 5
6 4 × 3 11 × 10 6 × 7 12 × 3 8 × 14
7 7 × 8 8 × 9 3 × 11 14 × 4 9 × 7
8 9 × 6 5 × 13 8 × 4 13 × 3 12 × 15
9 12 × 5 3 × 14 9 × 6 7 × 6 3 × 3
10 13 × 14 2 × 12 12 × 4 8 × 7 4 × 2
11 4 × 9 4 × 6 8 × 2 3 × 8 14 × 8
12 2 × 5 9 × 5 3 × 13 11 × 12 6 × 10
Dividing and multiplying numbers
Long multiplicationRapid multiplication of multiple numbers is easy if you know the
multiplication tables inside-out and back-to-front. In a long multipli-
cation calculation, you break the problem down into a number of
simple calculations by dividing the multiplier up into units of tens,
hundreds, thousands and so on. In Chapter 2 you will work through
practice drills involving division and multiplication of decimals.
Worked example
Q. What is the result of 2,348 s 237?
To multiply a number by 237, break the problem down into a number
of simpler calculations. Divide the multiplier up into units of hun-
dreds, tens and units.
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REVIEW THE BASICS 13
For example, to multiply by 237 you multiply by:
7 (units)
3 (tens)
2 (hundreds)
(It doesn’t matter in which order you complete the calculation.)
2,348
237 s Multiplier
16,436 2,348 s 7
70,440 2,348 s 30
469,600 2,348 s 200
556,476 16,436 70,440 469,600
Long multiplication: practice drill 1
No calculators! This exercise is intended to help you to speed up
your mental arithmetic. Set a stopwatch and aim to complete this
drill in five minutes.
Q1 12 × 24
Q2 13 × 23
Q3 11 × 23
Q4 19 × 19
Q5 26 × 24
Q6 213 × 43
Q7 342 × 45
Q8 438 × 23
Q9 539 × 125
Q10 5,478 × 762
Long multiplication: practice drill2
Set a stopwatch and aim to complete this practice drill in five
minutes.
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HOW TO PASS NUMERICAL REASONING TESTS14
Q1 9 × 18
Q2 11 × 19
Q3 12 × 21
Q4 19 × 23
Q5 26 × 19
Q6 211 × 17
Q7 317 × 13
Q8 416 × 11
Q9 624 × 97
Q10 725 × 101
Long division
Long division calculations, like long multiplication calculations, can
be completed quickly and easily without a calculator if you know the
multiplication tables well. There are four steps in a long division
calculation, and as long as you follow these in order, you will arrive
at the right answer.
Worked example
Q: Divide 156 by 12
This may seem obvious, but recognize which number you are divid-
ing into. This is called the dividend. In this case you are dividing the
dividend (156) by the divisor (12). Be clear about which is the divisor
and which is the dividend this will become very important when
you divide very large or very small numbers.
There are four steps in a long division question.
Step 1 Divide (D)
Step 2 Multiply (M)
Step 3 Subtract (S)
Step 4 Bring down (B)
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REVIEW THE BASICS 15
You can remember this as D-M-S-B with any mnemonic that
helps you to remember the order. Do-Mind-Slippery-Bananas.
Dirty-Muddy-Salty-Bicycles. Follow the steps in order and repeat
until you have worked through the whole calculation.
Step 1: Divide
Work from the left to the right of the whole number. 12 divides into
15 once, so write ‘1’ on top of the division bar.
12 1561
Step 2: MultiplyMultiply the result of step 1 (1) by the divisor (12):
1 s 12 12. Write the number 12 directly under the dividend
(156).
12 15612
1
Step 3: Subtract
Subtract 12 from 15 and write the result directly under the result of
Step 2.
12 156
1
12
3
Step 4: Bring down
Bring down the next digit of the dividend (6).
12 156
1
12
36
Return to Step 1 and start the four-step process again.
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HOW TO PASS NUMERICAL REASONING TESTS16
Step 1: Divide 12 into 36 and write the result (3) on top of the long
division sign.
12 156
13
12
36
Step 2: Multiply the result of Step 1 (3) by the divisor (12): 3 s 12
36. Write the number 36 directly below the new dividend (36).
12 156
13
12
36
36
Step 3: Subtract 36 from 36.
12 156
13
12
36
360
Step 4: There aren’t any more digits to Bring down, so the calcula-
tion is complete.
156 w 12 13
You will learn about long division with remainders and decimals in
Chapter 2.
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REVIEW THE BASICS 17
Long division: practice drill 1
Set a stopwatch and aim to complete these calculations in four
minutes. You may check your answers with a calculator only once
you have finished all the questions in the drill.
Q1 99 ÷ 11
Q2 91 ÷ 13
Q3 117 ÷ 13
Q4 182 ÷ 14
Q5 696 ÷ 58
Q6 3,024 ÷ 27
Q7 2,890 ÷ 34
Q8 636 ÷ 53
Q9 1,456 ÷ 13
Q10 2,496 ÷ 78
Long division: practice drill 2
Set a stopwatch and aim to complete the following practice drill
within five minutes.
Q1 1,288 ÷ 56
Q2 1,035 ÷ 45
Q3 6,328 ÷ 56
Q4 5,625 ÷ 125
Q5 2,142 ÷ 17
Q6 7,952 ÷ 142
Q7 10,626 ÷ 231
Q8 11,908 ÷ 458
Q9 81,685 ÷ 961
Q10 3,591 ÷ 27
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HOW TO PASS NUMERICAL REASONING TESTS18
Prime numbers An integer greater than 1 is a prime number if its only positive
divisors are itself and 1. All prime numbers apart from 2 are odd
numbers. Even numbers are divisible by 2 and cannot be prime by
definition. 1 is not a prime number, because it is divisible by one
number only, itself. The following is a list of all the prime numbers
below 100. It’s worth becoming familiar with these numbers so
that when you come across them in your test, you don’t waste time
trying to find other numbers to divide into them!
0
10 2 3 5 71120 11 13 17 19
2130 23 29
3140 31 37
4150 41 43 47
5160 53 59
6170 61 67
7180 71 73 79
8190 83 89
91100 97
Prime numbers: practice drill
Refer to the table above to assist you with the following drill:
Q1 What is the product of the first four prime numbers?
Q2 What is the sum of the prime numbers between 40 and 50
minus the eleventh prime number?
Q3 How many prime numbers are there?
Q4 How many prime numbers are there between 1 and 100?
Q5 What is the only even prime number?
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REVIEW THE BASICS 19
Q6 Between 1 and 100, there are five prime numbers ending
in 1. What are they?
Q7 What is the result of the product of the first three prime
numbers minus the sum of the second three prime numbers?
Q8 How many prime numbers are there between 60 and 80?
Q9 How many prime numbers are there between 90 and 100?
Q10 What is the sum of the second 12 prime numbers minus
the sum of the first 12 prime numbers?
Multiples
A multiple is a number that divides by another without a remainder.
For example, 54 is a multiple of 9 and 72 is a multiple of 8.
T ips to find multiples
An integer is divisible by:
2, if the last digit is 0 or is an even number
3, if the sum of its digits are a multiple of 34, if the last two digits are a multiple of 4
5, if the last digit is 0 or 5
6, if it is divisible by 2 and 3
9, if its digits sum to a multiple of 9
There is no consistent rule to find multiples of 7 or 8.
Worked example
Is 2,648 divisible by 2? Yes, because 8 is divisible by an even
number.
Is 91,542 divisible by 3? Yes, because 91542 21 and 21 is a
multiple of 3.
Is 216 divisible by 4? Yes, because 16 is a multiple of 4.
Is 36,545 divisible by 5? Yes, because the last digit is 5.
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HOW TO PASS NUMERICAL REASONING TESTS20
Is 9,918 divisible by 6? Yes, because the last digit, 8, is divisible
by an even number and the sum of all the digits, 27, is a
multiple of 3.
Multiples: practice drill
Set a stopwatch and aim to complete the following 10-question drill
in five minutes.
The following numbers are multiples of which of the following
integers: 2, 3, 4, 5, 6, 9?
Drill 1 Drill 2 Drill 3 Drill 4
1 36 2,654 642 5422 218 23 8,613 9,768
3 5,244 96 989,136 8,752
4 760 524 652 92
5 7,735 97 1,722 762
6 29 152 13 276
7 240,702 17,625 675 136
8 81,070 7,512 124 19
9 60,472 64 86 9005
10 161,174 128 93 65
Lowest common multiple
The lowest common multiple is the least quantity that is a multiple of
two or more given values. To find a multiple of two integers, you can
simply multiply them together, but this will not necessarily give you
the lowest common multiple of both integers. To find the lowest
common multiple, you will work with the prime numbers. This is a
concept you will find useful when working with fractions. There are
three steps to find the lowest common multiple of two or more
numbers:
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REVIEW THE BASICS 21
Step 1: Express each of the integers as the product of its prime
factors.
Step 2: Line up common prime factors.
Step 3: Find the product of the distinct prime factors.
Worked example
What is the lowest common multiple of 6 and 9?
Step 1: Express each of the integers as the product of its
prime factors
To find the prime factors of an integer, divide that number by the
prime numbers, starting with 2. The product of the prime factors ofan integer is called the prime factorization.
Divide 6 by 2:
2 6
3
Now divide the remainder, 3, by the next prime factor after 2:
3 3
1
So the prime factors of 6 are 2 and 3. (Remember that 1 is not a prime
number.) The product of the prime factors of an integer is called the
prime factorization, so the prime factorization of 6 2 s 3.
Now follow the same process to work out the prime factorization
of 9 by the same process. Divide 9 by the first prime number that
divides without a remainder:
3 9
3
Now divide the result by the first prime number that divides without
a remainder.
3 31
The prime factorization of 9 3 s 3.
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HOW TO PASS NUMERICAL REASONING TESTS22
Step 2: Line up common prime factors
Line up the prime factors of each of the given integers below each
other:
6 2 s 3
9 3 s 3
Notice that 6 and 9 have a common prime factor (3).
Step 3: Find the product of the prime factors
Multiply all the prime factors together. When you see a common
prime factor, count this only once.
6
2s
39 3 s 3
Prime factorization =
2 × 3 × 3 18
The lowest common multiple of 6 and 9 18.
Lowest common multiple: practice drill 1
Set a stopwatch and aim to complete the following drill in four
minutes. Find the lowest common multiple of the following sets
of numbers:Q1 8 and 6
Q2 12 and 9
Q3 3 and 5
Q4 12 and 15
Q5 8 and 14
Q6 9 and 18
Q7 4 and 7
Q8 13 and 7Q9 12 and 26
Q10 7 and 15
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REVIEW THE BASICS 23
Lowest common multiple: practice drill 2
Set a stopwatch and aim to complete the following drill in four minutes.
Find the lowest common multiple of the following sets of numbers:
Q1 2 and 5
Q2 4 and 5
Q3 5 and 9
Q4 6 and 5
Q5 6 and 7
Q6 2 and 5 and 6
Q7 3 and 6 and 7
Q8 3 and 7 and 8
Q9 3 and 6 and 11
Q10 4 and 6 and 7
Working with large numbers
Test-writers sometimes set questions that ask you to perform an
operation on very large or very small numbers. This is a cruel test
trap, as it is easy to be confused by a large number of decimal
places or zeros. Operations on small and large numbers are dealt
with in Chapter 2. This section reminds you of some commonly
used terms and their equivalents.
Millions, billions and trillions
The meaning of notations such as millions, billions and trillions is
ambiguous. The terms vary and the UK definition of these terms
is different from the US definition. If you are taking a test developed
in the United States (such as the GMAT or GRE), make sure you
know the difference.
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HOW TO PASS NUMERICAL REASONING TESTS24
US definition
Million: 1,000,000 or ‘a thousand thousand’ (same as UK definition)
Billion: 1,000,000,000 or ‘a thousand million’
Trillion: 1,000,000,000,000 or ‘a thousand billion’
UK definition
Million: 1,000,000 or ‘a thousand thousand’ (same as US definition)
Billion: 1,000,000,000,000 or ‘a million million’
Trillion: 1,000,000,000,000,000,000 or ‘a million million million’
The US definitions are more commonly used now. If you are in doubt
and do not have the means to clarify which notation is being used,assume the US definition.
Multiplying large numbers
To multiply large numbers containing enough zeros to make you go
cross-eyed, follow these three steps:
Step 1: Multiply the digits greater than 0 together.
Step 2: Count up the number of zeros in each number.
Step 3: Add that number of zeros to the result of Step 1.
Worked example
What is the result of 2,000,000 s 2,000?
Step 1: Multiply the digits greater than 0 together
2 s 2 4
Step 2: Count up the number of zeros in each number
2,000,000 s 2,000 nine zeros
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REVIEW THE BASICS 25
Step 3: Add that number of zeros to the right of the result of
Step 1
Result of Step 1 4
nine zeros 000,000,000
Answer 4,000,000,000 (4 billion, US definition)
Multiplying large numbers: practice drill
Use the US definition of billion and trillion to complete this practice
drill. Set a stopwatch and aim to complete the following drill in three
minutes.
Q1 9 hundred × 2 thousand
Q2 2 million × 3 million
Q3 3 billion × 1 million
Q4 12 thousand × 4 million
Q5 24 million × 2 billion
Q6 18 thousand × 2 million
Q7 2 thousand × 13 million
Q8 3 hundred thousand × 22 thousand
Q9 28 million × 12 thousand
Q10 14 billion × 6 thousand
Dividing large numbers
Divide large numbers in exactly the same way as you would smaller
numbers, but cancel out equivalent zeros before you start.
Worked example
4,000,000 w 2,000
Cancel out equivalent zeros:
4 000 000 2 000, , ,w
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HOW TO PASS NUMERICAL REASONING TESTS26
Now you are left with an easier calculation:
4,000 w 2
Answer 2,000
Dividing large numbers: practice drill
Set a stopwatch and aim to complete the following drill in four
minutes.
Q1 8,000 ÷ 20
Q2 2,700 ÷ 90
Q3 240,000 ÷ 600
Q4 6,720,000 ÷ 5,600
Q5 475,000 ÷ 1,900
Q6 19,500,000 ÷ 15,000
Q7 23,800,000 ÷ 140
Q8 149,500,000 ÷ 65,000
Q9 9,890,000 ÷ 2,300
Q10 15,540,000 ÷ 42,000
Working with signed numbers
Multiplication of signed numbers
There are a few simple rules to remember when multiplying signed
numbers.
Positive s positive positive P s P P
Negative s negative positive N s N P
Negative s positive negative N s P N
Positives
negative
negative Ps
N
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REVIEW THE BASICS 27
The product of an odd number of negatives negative
N s N s N N
The product of an even number of negatives positive
N s N s N s N P
Worked exampleP s P P 2 s 2 4
N s N P 2 s 2 4
N s P N 2 s 2 4
P s N N 2 s 2 4
N s N s N N 2 s 2 s 2 8
N s N s N s N P 2 s 2 s 2 s 2 16
Division of signed numbers
Positive w positive positive P w P P
Negative w negative positive N w N PNegative w positive negative N w P N
Positive w negative negative P w N N
Worked example
P w P P 2 w 2 1
N w N P 2 w 2 1
P w N N 2 w 2 1
N w P N 2 w 2 1
Tip: note that it doesn’t matter which sign is presented first in a multi-
plication calculation.
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HOW TO PASS NUMERICAL REASONING TESTS28
Multiplication and division of signed numbers:practice drill I
Set a stopwatch and aim to complete each drill within five minutes.
Q1 12 × 12
Q2 12 × 12
Q3 14 × 3
Q4 27 × 13
Q5 19 × 19
Q6 189 ÷ 21
Q7 84 ÷ 6Q8 1,440 ÷ 32
Q9 221 ÷ 17
Q10 414 ÷ 23
Multiplication and division of signed numbers: practicedrill2
Q1 16 × 13
Q2 27 × 29
Q3 131 × 21
Q4 52 × 136
Q5 272 × 13
Q6 112 ÷ 2
Q7 72 ÷ 24
Q8 540 ÷ 12
Q9 4,275 ÷ 19
Q10 3,638 ÷ 214
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REVIEW THE BASICS 29
AveragesOne way to compare sets of numbers presented in tables, graphs
or charts is by working out the average. This is a technique used in
statistical analysis to analyse data and to draw conclusions about
the content of the data set. The three types of averages are the
arithmetic mean, the mode and the median.
Arithmetic mean
The arithmetic mean (also known simply as the average) is a term
you are probably familiar with. To find the mean, simply add up allthe numbers in the set and divide by the number of terms.
Arithmetic meanSum of values
Number of values
Worked example
In her aptitude test, Emma scores 77, 81 and 82 in each section.
What is her average (arithmetic mean) score?
Arithmetic mean ( )77 81 82 240
3
Arithmetic mean 80
Worked example
What is the arithmetic mean of the following set of numbers: 0, 6,
12, 18?
Arithmetic mean ( )0 6 12 18 36
4
The arithmetic mean of the set is 36 w 4 9.
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HOW TO PASS NUMERICAL REASONING TESTS30
Worked example
What is the value of q if the arithmetic mean of 3, 6, 9 and q 5.5?
Step 1: Rearrange the formula to help you to find the sum of
values
Sum of values arithmetic mean s number of values
Step 2: Now plug in the numbers
Sum of values arithmetic mean s number of values
Sum of values 5.5 s 4 22
(Don’t forget to count the fourth value q in the number of values.)
Sum of values 22
Step 3: Subtract the sum of known values from the sum of
values
Sum of values sum of known values q
22 18 (3 6 9) q
Answer : q 4
The mode
The mode is the number (or numbers) that appear(s) the most
frequently in a set of numbers. There may be more than one mode
in a given set of numbers.
Tip: remember to include the zero as a value in your sum of the
number of values.
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REVIEW THE BASICS 31
Worked example
What is the mode in the following set of numbers?
21, 22, 23, 22, 22, 25, 25, 22, 21
21 appears twice.
22 appears four times.
23 appears once.
25 appears twice.
So the mode is 22 as it appears most frequently.
Worked example
What is the mode in the following set of numbers?
0.01, 0.01, 0.01, 0.1, 0.01, 0.01, 0.1, 0.01, 0.1
0.1 appears once.
0.01 appears three times.
0.1 appears twice.
0.01 appears three times.
So the mode numbers are 0.01 and 0.01.
The median
The median is the value of the middle number in a set of numbers,
when the numbers are put in ascending or descending order.
Worked example
What is the median in the following set of numbers?
82, 21, 34, 23, 12, 46, 65, 45, 37
First order the numbers in ascending (or descending) order.
12, 21, 23, 34, 37, 45, 46, 65, 82
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HOW TO PASS NUMERICAL REASONING TESTS32
As there are nine numbers in the set, the fifth number in the series is
the median.
1
12
2
21
3
23
4
34
5
6
45
7
46
8
65
9
82
37 is the median value.
Worked example
What is the median in the following set of numbers?
0, 2, 6, 10, 4, 8, 0, 1
First put the numbers of the set in order (either ascending or
descending).
0, 0, 1, 2, 4, 6, 8, 10
This time there is an even number of values in the set.
1
0
2
0
3
1
4
2
5
4
6
6
7
8
8
10
Draw a line in the middle of the set:
1
0
2
0
3
1
4
2
5
4
6
6
7
8
8
10
The median of the series is the average of the two numbers on either
side of the dividing line. Therefore, the median number in the series
is the arithmetic mean of 2 and 4:
(2 4) w 2 3.
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REVIEW THE BASICS 33
Averages: practice drill
Set a stopwatch and aim to complete the following drill in 60 seconds.
What is the arithmetic mean of the following sets of numbers?
Q1 17, 18, 21, 23, 21
Q2 0.1, 0.2, 0.2, 0.3, 0.3, 0.1
Q3 0, 2, 4, 6
Q4 2, 4, 6, 10
Q5 0.2, 0.1, 0.3, 0.6, 0.7, 0.9
What is the median of the following sets of numbers?
Q6 8, 1, 6, 2, 4
Q7 6, 2.5, 0, 1, 3
Q8 82, 73, 72, 72, 71
Q9 6, 0, 3, 9, 15, 12
Q10 36, 32, 37, 41, 39, 39
What is the mode in the following sets of numbers?
Q11 21, 22, 23, 23, 22, 23
Q12 2, 7, 2, 7, 7, 4, 4, 2, 3, 2
Q13 0, 1, 1, 1, 0, 0, 0, 1, 1
Q14 1
2
1
4
3
4
3
4
1
4
1
2
1
4, , , , , ,
Q15 0.1, 0.001, 0.01, 0.1, 0.001, 0.1, 0.001
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HOW TO PASS NUMERICAL REASONING TESTS34
Answers to Chapter 1Multiplication tables: practice drill 1
Drill 1 Drill 2 Drill 3 Drill 4 Drill 5
1 21 24 27 35 105
2 30 66 18 18 121
3 72 26 49 28 48
4 9 143 84 12 80
5 108 117 48 120 52
6 8 84 42 64 22
7 40 45 65 12 84
8 39 72 52 42 135
9 42 20 64 48 81
10 14 18 91 70 24
11 84 48 90 33 9
12 24 77 72 54 24
Multiplication tables: practice drill 2
Drill 1 Drill 2 Drill 3 Drill 4 Drill 5
1 72 20 18 66 121
2 169 27 45 15 21
3 154 52 21 30 48
4 54 48 45 40 48
5 55 49 165 196 25
6 12 110 42 36 112
7 56 72 33 56 63
8 54 65 32 39 180
9 60 42 54 42 9
10 182 24 48 56 8
11 36 24 16 24 112
12 10 45 39 132 60
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REVIEW THE BASICS 35
Long multiplication: practice drill 1
Q1 288
Q2 299
Q3 253
Q4 361
Q5 624
Q6 9,159
Q7 15,390
Q8 10,074
Q9 67,375
Q10 4,174,236
Long multiplication: practice drill 2
Q1 162
Q2 209
Q3 252Q4 437
Q5 494
Q6 3,587
Q7 4,121
Q8 4,576
Q9 60,528
Q10 73,225
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HOW TO PASS NUMERICAL REASONING TESTS36
Long division practice: drill 1
Q1 9
Q2 7
Q3 9
Q4 13
Q5 12
Q6 112
Q7 85
Q8 12
Q9 112
Q10 32
Long division practice: drill 2
Q1 23
Q2 23
Q3 113Q4 45
Q5 126
Q6 56
Q7 46
Q8 26
Q9 85
Q10 133
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REVIEW THE BASICS 37
Prime numbers: practice drill
Q1 210
Q2 100
Q3 An infinite number
Q4 25
Q5 2
Q6 11, 31, 41, 61, 71
Q7 1
Q8 5
Q9 1
Q10 569
Multiples: practice drill
Drill 1 Drill 2 Drill 3 Drill 4
1 2,3,4,6,9 2 2,3,6 2
2 2 Prime 3,9 2,3,4,6 [8]
3 2,3,4,6 2,3,4,6 2,3,4,6,9 [8] 2,4 [8]
4 2,4,5 2,4 2,4 2,4
5 5 [7] Prime 2,3,6 [7] 2,3,6
6 Prime 2,4 [8] Prime 2,3,4,6
7 2,3,6 3,5 3,5,9 2,4 [8]
8 2,5 2,3,4,6 [8] 2,4 Prime
9 2,4 [8] 2,4 [8] 2 5
10 2 2,4 [8] 3 5
Where the integer is also a multiple of 7 or 8, this is indicated in the
answer table with [7] or [8].
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HOW TO PASS NUMERICAL REASONING TESTS38
Lowest common multiple: practice drill 1
Q1 8 and 6
Answer 24
8 = 2 × 2 × 2
6 = 2 × 3
Prime factorization 2 s 2 s 2 s 3
Q2 12 and 9
Answer 36
12 = 2 × 2 × 3
9 = 3 × 3
Prime factorization 2 s 2 s 3 s 3
Q3 3 and 5
Answer 15
3 = 3
×
×
55 =
1
1
Prime factorization (1 s) 3 s 5
Q4 12 and 15
Answer 60
12 = 2 × 2 × 3
15 = 3 × 5
Prime factorization 2 s 2 s 3 s 5
Q5 8 and 14
Answer 56
8 = 2 × 2 × 2
14 = 2 × 7
Prime factorization 2 s 2 s 2 s 7
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REVIEW THE BASICS 39
Q6 9 and 18
Answer 18
9 = 3 ×
2 × 3 ×
3
18 = 3
Prime factorization 2 s 3 s 3
Q7 4 and 7
Answer 28
4 2 s 2
7 1 s 7
Prime factorization (1 s) 2 s 2 s 7
Q8 13 and 7
Answer 91
13 = 13
×
×
77 =
1
1
Prime factorization (1 s) 13 s 7
Q9 12 and 26
Answer 156
12 = 2 × 2 ×
×
3
26 = 132
Prime factorization 2 s 2 s 3 s 13
Q10 7 and 15
Answer 105
7 1 s 7
15 3 s 5
Prime factorization
(1s
) 3s
5s
7
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HOW TO PASS NUMERICAL REASONING TESTS40
Lowest common multiple: practice drill2
Q1 10
Q2 20
Q3 45
Q4 30
Q5 42
Q6 30
Q7 42
Q8 168
Q9 66
Q10 84
Multiplying large numbers: practice drill
Q1 1 million, 8 hundred thousand (1,800,000)
Q2 6 trillion (6,000,000,000,000)
Q3 3 thousand trillion (3,000,000,000,000,000)
Q4 48 billion (48,000,000,000)
Q5 48 thousand trillion (48,000,000,000,000,000)
Q6 36 billion (36,000,000,000)
Q7 26 billion (26,000,000,000)
Q8 6 billion, 6 hundred million (6,600,000,000)
Q9 336 billion (336,000,000,000)
Q10 84 trillion (84,000,000,000,000)
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REVIEW THE BASICS 41
Dividing large numbers: practice drill
Q1 400
Q2 30
Q3 400
Q4 1,200
Q5 250
Q6 1,300
Q7 170,000
Q8 2,300
Q9 4,300
Q10 370
Multiplication and division of signed numbers: practicedrill1
Q1 144
Q2 144
Q3 42
Q4 351Q5 361
Q6 9
Q7 14
Q8 45
Q9 13
Q10 18
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HOW TO PASS NUMERICAL REASONING TESTS42
Multiplication and division of signed numbers: practicedrill2
Q1 208
Q2 783
Q3 2,751
Q4 7,072
Q5 3,536
Q6 56
Q7 3
Q8 45
Q9 225
Q10 17
Averages: practice drill
Arithmetic mean:
Q1 20
Q2 0.2
Q3 3
Q4 5.5
Q5 0.4
Median:
Q6 4
Q7 2.5
Q8 72
Q9 7.5
Q10 38
Mode:
Q11 23
Q12 2
Q13 1
Q14
1
4Q15 0.1 and 0.001
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